Shifting Constraints in the Particle in a Box System

[Domnu]

I was just wondering... if a problem involved a particle which was constrained to move from x = -a/2 to a/2 and asked you to find it's properties (not position, though), could you just "shift" the entire system from x = -a/2, a/2 to x = 0 to a?

Also, let's say that a question asked for the probability of the particle being present from x = -a/2 to a/10 (assuming the particle is constrained from -a/2 to a/2). Could we just shift the box to 0 to a and find the probability of the particle being present in the areas between 0 and a/10+a/2 = 3a/5 ?

This would be really useful, because I can still use the eigenstates of the energy function for a particle in the box scenario,

$$\phi_n = \sqrt{\frac{2}{a}}\sin \frac{n\pi x}{a}$$

By the way, this is all assuming that no potential energy is present in the system.

 

[lbrits]

Yes, the Schrödinger equation is invariant under coordinate changes $$x \to x + b$$.
You'll also find that if you do the transformation on your eigenstates, you end up with cosines, which are solutions to the SE with the new boundary condition.